145, 146] ZONAL HARMONICS. 399 



Considering this as a function of z let us develop by Taylor's 

 Theorem, 



103) ^=^_/)= 



, . f A 1 1 0V /1\ 



and since tor r = 0, -r = > = l\> 



> d r 



*> J-7 + <-0 



Now multiplying and dividing each term by r n + l , we find 



1 1 



105) ^ = i 

 where 



106) P =l, P 



This is the determination of the constant A, adopted by Legendre, 

 for the reason that, since by the binomial theorem, for r'<r ; and ft= 1, 



d fc.'i* , V] r \ A r 



it makes for every w, 



107) P.(l) = 1. 



The term n * is a spherical harmonic of degree (n + 1), and 

 the series 105) is convergent for r\ < r. In like manner if r 1 > r we find 



^Z 0*' I i*' , 7*' ^ 



In order to find P n as a polynomial in [i we may write ^ as 



and develop by the binomial theorem. 



109) 



5 = 



Developing the last factor, 



